Foundation of fractional Langevin equation: Harmonization of a many-body problem

In this study we derive a single-particle equation of motion, from first principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models and a relation derived by Kollmann.

Figure 1

(Color online) Illustration of the many-particle system studied here. (a) Particles interact via the two-body potential $\mathcal{V}\left(\left|x_n\left(t\right)-x_{n^{\prime }}\left(t\right)\right|\right)$ and cannot pass each other. (b) The harmonization procedure amounts to mapping the interacting particle system onto a harmonic chain in which the interactions are captured by the effective spring constant $\kappa$.

Figure 2

(Color online) Autocorrelation function obtained from molecular dynamics simulations of ${10}^4$ point particles on a ring (solid lines) with corresponding theoretical predictions (dashed lines). Upper (black) lines: tracer particle in an external harmonic potential with the Mittag-Leffler function (13) (parameters are $\rho =0.25$, $k_{B}T=1$, $\xi =0.5$, and $m{\omega }^2=0.0158$, resulting in $\tau \approx 501$). Lower (red) lines: interparticle distance correlation function with the prediction of Eq. (18) (parameters are $\rho =0.4$, $k_{B}T=1$, $\xi =0.5$, and $n_{\text{a}}-n_{\text{d}}=10$, giving ${\tau }_{\text{dist}}\approx 313$).